CHAPTER The Elastic Stress Field around a Crack Tip 3 Brittle fracture in a solid in the form of crack growth is governed by the stress field around the crack tip and by parameters that describe the resistance of the material to crack growth. Thus, the analysis of stresses near the crack tip constitutes an essen- tial part of fracture mechanics. For brittle materials exhibiting linear elastic behavior, methods of elasticity are used to obtain stresses and displacements in cracked bod- ies. These methods include analytical ones, such as the complex potential function method and the integral transform method, and numerical ones, such as the finite ele- ment method. In this chapter, the complex potential function method is introduced and used to analyze the stresses and displacements around crack tips. The characteri- stics of the near-tip asymptotic stress and displacement fields and the crack growth criterion based on the crack tip field are discussed. 3.1 BASIC MODES OF FRACTURE AND STRESS INTENSITY FACTOR A crack in a solid consists of disjoined upper and lower faces. The joint of the two crack faces forms the crack front. The two crack faces are usually assumed to lie in the same surface before deformation. When the cracked body is subjected to external loads (remotely or at the crack surfaces), the two crack faces move with respect to each other and these movements may be described by the differences in displace- ments ux, uy, and uz between the upper and lower crack surfaces, where .x,y,z/ is a local Cartesian coordinate system centered at the crack front with the x-axis perpen- dicular to the crack front, the y-axis perpendicular to the crack plane, and the z-axis along the crack front. There are three independent movements corresponding to three fundamental frac- ture modes as pointed out by Irwin [3-1], which are schematically illustrated in Figure 3.1. These basic fracture modes are usually called Mode I, Mode II, and Mode III, respectively, and any fracture mode in a cracked body may be described by one of the three basic modes, or their combinations (see their descriptions on the next page). Fracture Mechanics 25 c 2012 Elsevier Inc. All rights reserved. 26 CHAPTER 3 The Elastic Stress Field around a Crack Tip y y y x x x z z z (a) (b)(c) FIGURE 3.1 Schematic of the basic fracture modes: (a) Mode I (opening), (b) Mode II (sliding), (c) Mode III (tearing). 1. Mode I (Opening Mode): The two crack surfaces experience a jump only in uy, that is, they move away symmetrically with respect to the undeformed crack plane (xz-plane). 2. Mode II (Sliding Mode): The two crack surfaces experience a jump only in ux, that is, they slide against each other along directions perpendicular to the crack front but in the same undeformed plane. 3. Mode III (Tearing Mode): The two crack surfaces experience a jump only in uz, that is, they tear over each other in the directions parallel to the crack front but in the same undeformed plane. The three basic modes of crack deformation can be more precisely defined by the associated stresses ahead of the crack front, which may be considered as the crack tip in two-dimensional problems. It will be seen in the following sections that the near-tip stresses in the crack plane (xz-plane) for these three modes can be expressed as (y 0, x 0 ) D ! C KI σyy O.px/, σxy σyz 0 D p2πx C D D KII σxy O.px/, σyy σyz 0 (3.1) D p2πx C D D KIII σyz O.px/, σyy σxy 0 D p2πx C D D respectively, where the three parameters KI, KII, and KIII are named stress intensity factors corresponding to the opening, sliding, and tearing (anti-plane shearing) modes of fracture, respectively. These expressions show that the stresses have an inverse square root singularity at the crack tip and the stress intensity factors KI, KII, and KIII measure the intensities of the singular stress fields of opening, in-plane shearing, and anti-plane shearing, respectively. The stress intensity factor is a new concept in mechanics of solids and 3.2 Method of Complex Potential for Plane Elasticity 27 FIGURE 3.2 Mode III deformation in a cracked cylinder under torsion. plays an essential role in the study of fracture strength of cracked solids. Various methods for determining stress intensity factors, including analytical, numerical, and experimental approaches, have been developed in the past few decades. It is important to note that, except for Mode I deformation shown in Figure 3.1(a), the loading and specimen geometries shown in Figures 3.1(b) and (c) cannot be used to produce pure Mode II and Mode III deformation, respectively. In fact, unless an additional loading or boundary condition is specified, the cracked bodies cannot be in equilibrium. Other types of specimen are usually used. For instance, a long cylinder of a circular cross-section with a longitudinal slit under torsion (see Figure 3.2) can be used to produce a pure Mode III crack deformation. 3.2 METHOD OF COMPLEX POTENTIAL FOR PLANE ELASTICITY (THE KOLOSOV-MUSKHELISHVILI FORMULAS) Among various mathematical methods in plane elasticity, the complex potential function method by Kolosov and Muskhelishvili [3-2] are one of the powerful and convenient methods to treat two-dimensional crack problems. In the complex poten- tial method, stresses and displacements are expressed in terms of analytic functions of complex variables. The problem of obtaining stresses and displacements around a crack tip is converted to finding some analytic functions subjected to appropriate boundary conditions. A brief introduction of the general formulation of the Kolosov and Muskhelishvili complex potentials is given in this section. 3.2.1 Basic Equations of Plane Elasticity and Airy Stress Function The basic equations of elasticity consist of equilibrium equations of stresses, strain- displacement relations, and Hooke’s law that relates stresses and strains. In plane elasticity (plane strain and plane stress), the equilibrium equations are (body forces are absent) @σ @σxy xx 0 @x C @y D (3.2) @σxy @σyy 0 @x C @y D 28 CHAPTER 3 The Elastic Stress Field around a Crack Tip where σxx, σyy, and σxy are stresses, and (x,y) are Cartesian coordinates. The strains and the displacements are related by @ux @uy 1 @ux @uy exx , eyy , exy (3.3) D @x D @y D 2 @y C @x where exx, eyy, and exy are tensorial strain components, and ux and uy are displace- ments. The stress–strain relations are given by σxx λ∗ exx eyy 2µexx D C C σyy λ∗ exx eyy 2µeyy (3.4) D C C σxy 2µexy D or inversely 1 λ∗ exx σxx σxx σyy D 2µ − 2.λ µ/ C ∗ C 1 λ∗ eyy σyy σxx σyy (3.5) D 2µ − 2.λ µ/ C ∗ C 1 exy σxy D 2µ where µ is the shear modulus and 3 κ λ∗ − µ D κ 1 − in which 8 <3 4ν for plane strain κ 3− ν (3.6) D − for plane stress :1 ν C In the previous relation, ν is Poisson’s ratio. The compatibility equation of strains can be obtained from Eq. (3.3) by eliminating the displacements as follows: 2 2 2 @ e @ eyy @ exy xx 2 (3.7) @y2 C @x2 D @x@y By using the stress–strain relations Eq. (3.5) together with the equations of equilib- rium Eq. (3.2), the compatibility condition Eq. (3.7) can be expressed in terms of stresses as 2 (σxx σyy/ 0 (3.8) r C D 3.2 Method of Complex Potential for Plane Elasticity 29 where 2 2 2 @ @ r D @x2 C @y2 is the Laplace operator. The Airy stress function φ is defined through @2φ @2φ @2φ σxx , σxy , σyy (3.9) D @y2 D −@x@y D @x2 Using these relations, the equilibrium equations in Eq. (3.2) are automatically satis- fied, and the compatibility Eq. (3.8) becomes 4φ 2 2φ 0 (3.10) r D r r D where @4 @4 @4 4 2 2 2 r D r r D @x4 C @x2@y2 C @y4 is the biharmonic operator. Any function φ satisfying Eq. (3.10) is called a bihar- monic function. A harmonic function f satisfies 2f 0. Thus, if f is harmonic, it is also biharmonic. However, the converse is notr true.D Once the Airy stress function is known, the stresses can be obtained by Eq. (3.9) and strains and displacements obtained through Eqs. (3.5) and (3.3), respectively. 3.2.2 Analytic Functions and Cauchy-Riemann Equations In a Cartesian coordinate system (x,y/, the complex variable z and its conjugate z are defined as N z x iy D C and z x iy N D − respectively, where i p 1. They can also be expressed in polar coordinates .r,θ/ as D − z r.cosθ isinθ/ reiθ D C D and iθ z r.cosθ isinθ/ re− N D − D respectively. 30 CHAPTER 3 The Elastic Stress Field around a Crack Tip Consider a function of the complex variable z, f .z/. The derivative of f .z/ with respect to z is by definition df .z/ f .z 1z/ f .z/ lim C − dz D 1z 0 1z ! If f .z/ has a derivative at point z0 and also at each point in some neighborhood of z0, then f .z/ is said to be analytic at z0.